Riemann Surfaces

Vorlesung von O. Forster im SS 2016
am Mathematischen Institut der LMU München

Wed, Fri 14-16, HS A027, Theresienstr. 39

Exercises Wed 16-18 (A027)

What this course is about: Every serious study of analytic functions of one complex variable will need Riemann surfaces. For example, "multi-valued" functions like square root or logarithm can be treated in a satisfactory way using Riemann surfaces covering the complex plane. Abstractly speaking, a Riemann surface is simply a complex 1-dimensional manifold (which looks locally like an open set in the complex plane). This course gives an introduction to the theory of Riemann surfaces with special focus on compact Riemann surfaces.
Some topics treated in this course: Definitions and basic properties. Construction of Riemann surfaces associated to algebraic functions and to algebraic curves. Divisors, line bundles, Theorem of Riemann-Roch. Periods of differential forms, Abel's Theorem, Jacobi Inversion Problem.

für: Studierende der Mathematik und Theoretischen Physik im Hauptstudium
mit Interesse in Funktionentheorie, Algebraischer Geometrie oder Differentialgeometrie.

Vorkenntnisse: Vorlesung Funktionentheorie I.
Nützlich sind auch Grundkenntnisse aus Algebra, Topologie oder Differentialgeometrie.

Leistungsnachweis: Gilt für Masterstudiengang Mathematik (WP37 oder WP36, WP34), Masterstudiengang TMP

Contents:

  1. Definition of Riemann surfaces
  2. Elementary properties of holomorphic maps
  3. Branched and unbranched coverings
  4. Riemann surfaces of algebraic functions
  5. Sheaves
  6. Cohomology groups
  7. Theorem of Riemann-Roch
  8. The Serre Duality Theorem
  9. Harmonic Differential Forms
  10. Jacobi Variety and Abel's Theorem

Literature:

  1. S. Donaldson: Riemann surfaces. Oxford Univ. Press.
  2. Farkas/Kra: Riemann Surfaces. Springer
  3. O. Forster: Lectures on Riemann Surfaces. Springer
  4. Gunning: Lectures on Riemann Surfaces. Mathematical Notes. Princeton University Press
  5. J. Jost: Compact Riemann Surfaces. Springer
  6. K. Lamotke: Riemannsche Flächen. Springer

Otto Forster (email), 2012-01-18